Using [1]} we see that
\(\hat{R}\)  defines an involutive \(*\) -anti-automorphism on \(W^*(G)\)  and that \(\hat{S}(\mathcal {F}(G)) = \mathcal {F}(G)\)  (but \(\hat{S}(W^*(G)) \subseteq W^*(G)\)  does not hold in general). Moreover, both \(\hat{S}\)  and \(\hat{R}\)  send each direct summand \(z\,\mathcal {U}(G) = zW^*(G) \cong B(\mathcal {H}_{U_z})\)  to the one \(\bar{z}\,\mathcal {U}(G)=\bar{z}W^*(G)\cong B(\mathcal {H}_{U_{\bar{z}}})\)  by (REF ), where \(\bar{z} \in \mathfrak {Z}\)  is a unique element such that \(U_{\bar{z}}\)  is (unitarily) equivalent to the conjugate representation \(\bar{U}_z\)  of \(U_z\)  (see [1]}). In this way, we have an involutive bijection \(z \mapsto \bar{z}\)  on \(\mathfrak {Z}\)  so that
\(\hat{S}(\bar{z}) = \hat{R}(\bar{z}) = z \) 
